zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
High-precision computations of divergent asymptotic series and homoclinic phenomena. (English) Zbl 1169.37013
A method suitable for studying the exponentially small splitting of separatrices appearing in the generalizations of standard map: $$ x_{1}=x+y_{1},\quad y_{1}=y+\epsilon f(x), $$ with $f$ being a polynomial, trigonometric polynomial, meromorphic or rational function is developed. The method is a combination of analytical and numerical steps, with high-precision computations. After the introduction, the analytical results on the splitting of separatrices from the generalized standard map are reviewed. Then, full details of numerical methods sketched in {\it C. Simó} [in: International conference on differential equations. Proceedings of the conference, Equadiff ’99, Berlin, Germany, August 1--7, 1999. Vol. 2. Singapore: World Scientific. 967--976 (2000; Zbl 0963.65136)] are given. The numerical procedure consists of two main steps: first, the values of the homoclinic invariant are computed; then, the obtained data are used to extract coefficients of an asymptotic expansion. After that, asymptotic formulae for $f(x)$ being a polynomial of degree $2$ to $5$ are described in detail. Finally, singularities of the separatrix solutions of the ODE: $\ddot{x}_{0}=f(x_{0})$ are studied, both in the case when these solutions can be found explicitly and when this is not possible in terms of elementary functions.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
65P10Numerical methods for Hamiltonian systems including symplectic integrators
37C29Homoclinic and heteroclinic orbits
37G20Hyperbolic singular points with homoclinic trajectories
Full Text: DOI Link